Since we’re using the (very small) finite field $\mathbb Z_2$ then we can actually list out all the vectors in our chain (group) vector space. We have 3 chain groups, namely the group of 0-simplices (vertices), 1-simplices (edges), and 2-simplices (triangle).
In our example, we only have a single 2-simplex: [a,b,c], thus the group it generates over the field $\mathbb Z_2$ is only ${0, [a,b,c]}$ which is isomorphic to $\mathbb Z_2$. Recall, in general, the group generated by the number $n$ of $p$-simplices in a simplicial complex is isomorphic to $\mathbb Z^n_2$. For a computer to understand, we can encode the group elements just using their coefficients 1 or 1. So, for example, the group generated by $[a,b,c]$ can just be represented as ${0,1}$. Or the group generated by the 0-simplices ${a, b, c, d}$ can be represented by 4-dimensional vectors, for example, if a group element is $a+b+c$ then we encode this as $(1, 1, 1, 0)$ where each position represents the presence or abscence of $(a, b, c, d)$, respectively.
Here are all the chain groups represented as vectors with just coefficients (I didn’t list all elements for $C_1$ since there are so many [32]):